Question

You can use a right triangle with leg lengths a and b and hypotenuse length c to prove the Pythagorean Theorem. Show that a² + b² = c².
Step 1: Make a square with an area of c².
b
a
The unshaded shape has 4 sides of length c. What else must be true for the shape to be a square with an area of c²?
C
C
b
>>The unshaded shape must have
b
?
C
a
a
b

Answer 1

Answer: the unshaded shape must have all four interior angles equal to 90° (i.e. adjacent sides are perpendicular).

Reason: each corner of the unshaded quadrilateral is formed by two acute angles of the right triangles; those two acute angles sum to 90°, so every corner is a right angle. With four equal sides c and four right angles the unshaded shape is a square of side c, so its area is c².

Finish the proof: place the four right triangles (legs a, b; hypotenuse c) inside a big square of side a + b. The area of the big square is (a + b)². That equals the area of the four triangles plus the inner square:
(a + b)² = 4·(1/2·ab) + c² = 2ab + c².
Expand the left side: a² + 2ab + b² = 2ab + c². Cancel 2ab to get
a² + b² = c².